Contrast agent kinetics is the study of the change of contrast agent concentration as a function of time, as the contrast agent enters into and perfuses a tissue of interest. Contrast agent kinetic analysis has applications in medical diagnostics by helping to characterize the functional state of a tissue, and applications in drug discovery by offering insight into the behavior of the contrast agent itself. Contrast agent kinetic analysis is used often in conjunction with an imaging device that can measure non-invasively, the concentration of the contrast agent, at one or more locations, as a function of time.
The fundamental behavior of a contrast agent in a tissue is quantified by the tissue impulse response function, R(t). The tissue impulse response function R(t) represents the temporal evolution of contrast agent concentration within a tissue if an ideal bolus, a Dirac delta function, of contrast agent had entered into that tissue without delay as described by Bassingthwaighte et al. in “Modeling in the analysis of solute and water exchange in the microvasculature”, Handbook of Physiology Section 2: The Cardiovascular System ed Renkin E and Geiger S (Bethesda Md.: American Physiological Society) 549-626, 1984. The tissue impulse response function R(t) is a function of both the contrast agent properties and the underlying tissue properties, and can be used to calculate many useful parameters such as blood flow, volume of distribution, extraction fraction, mean transit time, and standard deviation of transit time as described by Gobbel et al. in “A deconvolution method for evaluating indicator-dilution curves”, Physics in Medicine and Biology, Vol. 39, 1833-1854, 1994. From basic physiologic considerations, the expected shape of the tissue impulse response function R(t) is zero until the contrast agent enters the tissue, at which point the tissue impulse response function R(t) achieves its maximum value and then remains constant at this maximum value until the contrast agent begins to leave the tissue. Finally, the tissue impulse response function R(t) returns to zero over a period of time determined by the distribution of transit times of the contrast agent molecules.
In practice, the tissue impulse response function R(t) cannot be observed directly because an ideal bolus of contrast agent cannot be directly injected into the tissue of interest. Rather, a non-ideal bolus of contrast agent is typically introduced into the blood at a location upstream of the tissue of interest. As the contrast agent moves through the blood, the contrast agent concentration is measured by an imaging device, as a function of time, in two locations, one location of which is after the location of injection but upstream of the tissue of interest, I(t), and the other location of which is within the tissue of interest J(t). J(t) represents a combination of both R(t) and I(t), namely the tissue impulse response function whose shape has been complicated by the non-ideality and delay of the bolus. Mathematically, J(t) is the convolution of R(t) and I(t). The effect of a non-ideal I(t) must be corrected in order to calculate R(t) as described by Zierler in “Equations for measuring blood flow by external monitoring of radioisotopes”, Circulation Research, Vol. 16, 309-321, 1965. This correction is known as deconvolution.
Given that I(t) and J(t) are both corrupted by noise, the tissue impulse response function can be only estimated as R′(t). There are three major strategies for calculating R′(t), namely Frequency Domain Strategy, Analytical Model Strategy, and Spatial-Domain Strategy. Deconvolution analysis is difficult because it is inherently a differentiating process that is mathematically unstable and therefore very susceptible to noise in either I(t) or J(t) as described by Bronikowski et al. in “Model-free deconvolution techniques for estimating vascular transport function”, International Journal of Biomedical Engineering and Technology, Vol. 14, 411-429, 1983. The three strategies have evolved from differing opinions regarding stability of numerical methods.
Deconvolving I(t) from J(t) in the time domain is a difficult and unstable process. Theoretically, deconvolution can be simplified by using Fourier Theory to transform the two functions into their frequency domain counterparts FI(v) and FT(v), where v denotes frequency. The frequency domain counterpart of R′(t), namely FR′(v), can be written as the ratio of FT(v) and FI(v), namely FR′(v)=FT(v)/FI(v). Finally, FR′(v) can be transformed back into the spatial domain to yield R′(t) as described by Alderson et al. in “Deconvolution analysis in radionuclide quantitation of left-to-right cardiac shunts”, Journal of Nuclear Medicine, Vol. 20, 502-506, 1979, by Gamel et al. in “Pitfalls in digital computation of the impulse response of vascular beds from indicator-dilution curves”, Circulation Research, Vol. 32, 516-523, 1973, by Kuruc et al. in “Accuracy of deconvolution algorithms assessed by simulation studies: Concise Communication”, Journal of Nuclear Medicine, Vol. 24, 258-263, 1983 and by Juni et al. in “The appended curve technique for deconvolution analysis—method and validation”, European Journal of Nuclear Medicine, Vol. 14, 403-407, 1988.
In practice, the Fast Fourier Transform (FFT) as described by Cooley et al. in “An algorithm for the machine calculation of complex Fourier series”, Mathematics of Computation, Vol. 19, 297-301, 1965, and the Inverse FFT are used to move data between the spatial and frequency domains. FFT algorithms assume that I(t) and J(t) are periodic and without discontinuities as described by Wall et al. in “System parameter identification in transport models using the fast Fourier Transform (FFT)”, Computers and Biomedical Research, Vol. 14, 570-581, 1980. Unfortunately, significant discontinuities appear if data acquisition is terminated early because of time limitations, while a significant amount of contrast agent remains within the tissue of interest and J(t) has not yet dropped to zero as described in the aforementioned Gobbel et al. reference. Such an abrupt end to J(t), coupled with inherently noisy measurements, produces high-frequency oscillations in the calculated R′(t), as described in the aforementioned Juni et al. reference, that have no underlying physiologic basis as described by Bronikowski et al. in “Model-free deconvolution techniques for estimating vascular transport function”, International Journal of Biomedical Engineering and Technology, Vol. 14, 411-429. To address this concern, some authors have developed techniques to temporally extrapolate I(t) and J(t) to zero as described in the aforementioned Gobbel et al., Juni et al. and Wall et al. references; however the validity domain of temporal extrapolations of data has not been fully explored.
The Analytical Model Strategy assumes that the tissue impulse response function can be described by an analytical model that is derived from assumptions about the behavior of the contrast agent within the biological system under investigation as described by Nakamura et al. in “Detection and quantitation of left-to-right shunts from radionuclide angiography using the hormomorphic deconvolution technique”, IEEE Biomedical Engineering, Vol. 23, 192-201, 1982, and in the aforementioned Kuruc et al. reference. The analytical model of the tissue impulse response function has parameters that are initially unknown and must be calculated by fitting the model to the data. In general, reducing the number of parameters improves the stability of the model in the presence of noise. Reducing the number of parameters can however, limit the model as the final shape of the tissue impulse response function is restricted to the family of curves that can be realized by manipulating the parameters of the analytical model. The Analytical Model Strategy is a good choice for situations where an analytical model is known to characterize adequately healthy and pathologic tissue. Analytical models work best in specific well-defined biological systems, and should not be used beyond such systems since it is difficult to know a priori the appropriate functional form as described in the aforementioned Juni et al. reference.
The Matrix Strategy is based on an equivalency between discrete convolution and matrix multiplication. The expression J(t)=I(t)R′(t) can be re-written as MJ=MI×MR′, where ×denotes matrix multiplication; MJ and MR′ are vectors representing the I(t) and R′(t) data; and MI is a lower-triangular Toeplitz matrix of the I(t) values. Thus, MR′ can be expressed as the solution to this linear set of equations as described by Valentinuzzi et al. in “Discrete deconvolution”, Engineering in Medicine and Biology, Vol. 13, 123-125, 1975, by Ham et al. in “Radionuclide quantitation to left-to-right cardiac shunts using deconvolution analysis: Concise Communication”, Journal of Nuclear Medicine, Vol. 22, 688-692, 1981, and by Cosgriff et al. in “A comparative assessment of deconvolution and diuresis renography in equivocal upper urinary tract obstructions”, Nuclear Medicine Communications, Vol. 3, 377-384, 1982.
The Matrix Strategy has the advantages of neither requiring assumptions of an underlying analytical model nor requiring curves to go to zero. Unfortunately, this strategy can produce widely-oscillating R′(t) that have no physiological significance. To minimize these effects, I(t) and/or J(t) can be pre-processed or R′(t) can be post-processed to reduce their oscillatory nature as described by Basic et al. in “Extravascular background subtraction using deconvolution analysis of the renogram”, Physics in Medicine and Biology, Vol. 33, 1065-1073, 1988, and by Gonzalez et al. in “99Tcm-MAG3 renogram deconvolution in normal subjects and in normal functioning kidney grafts”, Nuclear Medicine Communications, Vol. 15, 680-684, 1994. Furthermore, R′(t) derived from Matrix Strategy methods can lead to R′(t) having values less than zero, which have no physical significance and must be set to zero as described in the aforementioned Gonzalez et al. reference.
Other contrast agent measurement techniques have also been considered. For example, U.S. Pat. No. 5,135,000 to Akselrod et al. discloses a method of measuring blood flow through tissue in a region of interest. The method includes the steps of injecting an ultrasonic tracer into the blood upstream of the region of interest and also upstream of a specified reference region; utilizing the tracer-produced echo intensity function to compute: (i) the mean transit time of the tracer through the region of interest; and (ii) the blood volume within the region of interest; and dividing the results of computation (ii) by the results of computation (i) to produce a quantitative measurement of the blood flow through the tissue in the region of interest.
The publication entitled “Myocardial Blood Flow Quantification with MRI By Model-Independent Deconvolution” authored by Jerosch-Herold et al. (Physics in Medicine 2002) discloses a method for determining blood flow quantification using model-free deconvolution, the Matrix Strategy and Tikhonov Regularization. The tissue impulse response function is constrained to be a sequence of B-splines (smoothly varying linked polynomial curves). The Matrix Strategy that is employed solves directly for the B-spline coefficients.
U.S. Pat. No. 6,542,769 to Schwamm et al. discloses an imaging system and method for obtaining quantitative perfusion indices. A bolus containing optical and MRI contrast agents is administered to a patient for determining quantitative perfusion indices from perfusion weighted magnetic resonance imaging analysis (PWI). The optical contrast agent time-concentration data, which can be obtained non-invasively, is used to define an arterial input function. The MRI contrast agent time concentration can be non-invasively determined using MRI to define a tissue function. An MRI time-signal curve can be determined by deconvolving the arterial and tissue functions. In one embodiment, SVD is used to determine a residue function from which a flow map can be computed.
U.S. Pat. No. 6,898,453 to Lee discloses a method for determining tissue type using blood flow and contrast agent transit time. Tissue blood flow (TBF) is determined by deconvoluting Q(t) and Ca(t), where Q(t) represents a curve of specific mass of contrast, and Ca(t) represents an arterial curve of contrast concentration, and quantitatively determining a tissue blood volume (TBV) by deconvoluting Q(t) and Ca(t). The method also includes quantitatively determining a tissue mean transit time (TMTT) by deconvoluting Q(t) and Ca(t), and quantitatively determining a tissue capillary permeability surface area product (TPS) by deconvoluting Q(t) and Ca(t). The method also includes determining a tissue type based on the TBF, the TBV, the TMTT, and the TPS.
U.S. Pat. No. 5,287,273 to Kupfer et al. discloses a noninvasive method of determining function of a target organ using a pre-calibrated imaging system. The method contains the steps of introducing an indicator/tracer bolus into the subject's circulatory system and thereafter monitoring simultaneously the responses recorded from the heart/great vessels, and from the target organ. The absolute activity per unit volume of blood withdrawn at a known time(s) is measured, and the observed data from the heart or great vessels is converted into absolute units. These data serve as B(t), the input function. Precalibration of the detector/measuring system allows the observed dynamic indicator/tracer data recorded from the target organ to be expressed in units of absolute activity. These data serve as A(t). A(t) and B(t) are deconvolved in order to obtain the linear response function (LRF), h(t)) for an image element. Functional images of the target organ's LRF are created.
Although contrast measurement techniques exist as is apparent from the above discussion, improvements are desired. It is therefore an object of the present invention at least to provide a novel method and apparatus for quantifying the behaviour of a contrast agent administered to a subject.